NTA JEE Main 20th July 2021 Shift 2

Let $$f : R - \{\frac{\alpha}{6}\} \to R$$ be defined by $$f(x) = \left(\frac{5x+3}{6x-\alpha}\right)$$. Then the value of $$\alpha$$ for which $$(f \circ f)(x) = x$$, for all $$x \in R - \{\frac{\alpha}{6}\}$$, is:

The sum of all the local minimum values of the twice differentiable function $$f : R \to R$$ defined by $$f(x) = x^3 - 3x^2 - \frac{3f''(2)}{2}x + f''(1)$$ is:

If $$[x]$$ denotes the greatest integer less than or equal to $$x$$, then the value of the integral $$\int_{-\pi/2}^{\pi/2} [x] - \sin x] dx$$ is equal to:

If $$f : R \to R$$ is given by $$f(x) = x + 1$$, then the value of
$$\lim_{n \to \infty} \frac{1}{n}\left[f(0) + f\left(\frac{5}{n}\right) + f\left(\frac{10}{n}\right) + \ldots + f\left(\frac{5(n-1)}{n}\right)\right]$$ is:

Let $$g(t) = \int_{-\pi/2}^{\pi/2} (\cos \frac{\pi}{4}t + f(x))dx$$, where $$f(x) = \log_e(x + \sqrt{x^2+1})$$, $$x \in R$$. Then which one of the following is correct?

Let $$y = y(x)$$ satisfies the equation $$\frac{dy}{dx} - |A| = 0$$, for all $$x > 0$$, where $$A = \begin{bmatrix} y & \sin x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x} \end{bmatrix}$$. If $$y(\pi) = \pi + 2$$, then the value of $$y\left(\frac{\pi}{2}\right)$$ is:

In a triangle $$ABC$$, if $$|\overrightarrow{BC}| = 3$$, $$|\overrightarrow{CA}| = 5$$ and $$|\overrightarrow{BA}| = 7$$, then the projection of the vector $$\overrightarrow{BA}$$ on $$\overrightarrow{BC}$$ is equal to:

The lines $$x = ay - 1 = z - 2$$ and $$x = 3y - 2 = bz - 2$$, $$(ab \neq 0)$$ are coplanar, if:

Consider the line $$L$$ given by the equation $$\frac{x-3}{2} = \frac{y-1}{1} = \frac{z-2}{1}$$. Let $$Q$$ be the mirror image of the point $$(2, 3, -1)$$ with respect to $$L$$. Let a plane $$P$$ be such that it passes through $$Q$$, and the line $$L$$ is perpendicular to $$P$$. Then which of the following points is on the plane $$P$$?

Let $$A$$, $$B$$, $$C$$ be three events such that the probability that exactly one of $$A$$ and $$B$$ occurs is $$(1-k)$$, the probability that exactly one of $$B$$ and $$C$$ occurs is $$(1-2k)$$, the probability that exactly one of $$C$$ and $$A$$ occurs is $$(1-k)$$ and the probability of all $$A$$, $$B$$ and $$C$$ occur simultaneously is $$k^2$$, where $$0 < k < 1$$. Then the probability that at least one of $$A$$, $$B$$ and $$C$$ occurs is: